Solving the supervised learning problem of learning $p(y \vert \mathbf{x})$ by using traditional unsupervised technologies to learn $p(\mathbf{x}, y)$
This is the definition of conditional probability + Total probability decomposition formula:

$p(y|x) = \frac{p(y,x}{p(x)} = \frac{p(x,y)}{\sum_{y'}p(x,y')}$. 

The idea is to use some unsupervised learning algorithm to learn the distribution $p(x,y)$ for every possible value of $y$, and by using the previous formula you can find $p(y|x)$.